Editing Network analysis (electrical circuits) (section)

===Delta-wye transformation===
{{main|Y-Δ transform}}

[[Image:Delta-Star Transformation.svg|right|400px]]

A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit.  An {{mvar|n}}-terminal network can, at best, be reduced to {{mvar|n}} impedances (at worst [[Binomial coefficient|<math>\tbinom{n}{2}</math>]]).  For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or  four node star (Y) network.  These two networks are equivalent and the transformations between them are given below.  A general network with an arbitrary number of nodes cannot be reduced to the minimum number of impedances using only series and parallel combinations.  In general, Y-Δ and Δ-Y transformations must also be used.  For some networks the extension of Y-Δ to [[#General form of network node elimination|star-polygon]] transformations may also be required.

For equivalence, the impedances between any pair of terminals must be the same for both networks, resulting in a set of three simultaneous equations.  The equations below are expressed as resistances but apply equally to the general case with impedances.

====Delta-to-star transformation equations====
:<math>\begin{align}
R_a &=  \frac{R_\mathrm{ac}R_\mathrm{ab}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} \\
R_b &=  \frac{R_\mathrm{ab}R_\mathrm{bc}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} \\
R_c &=  \frac{R_\mathrm{bc}R_\mathrm{ac}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} 
\end{align}</math>

====Star-to-delta transformation equations====

:<math>\begin{align}
R_\mathrm{ac} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_b} \\
R_\mathrm{ab} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_c} \\
R_\mathrm{bc} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_a}
\end{align}</math>